Five-Dimensional Fission-Barrier Calulations from Se-70 to Cf-252

0.17 MB·English

by  Peter Moller

Five-Dimensional Fission-Barrier Calculations from 70Se to 252Cf
Peter MO¨LLER1,∗ Arnold J. SIERK1, and Akira IWAMOTO2
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki, 319-1195 Japan
(Dated: February 8, 2008)
Wepresentfission-barrier-heightcalculationsfornucleithroughoutthePeriodicTablebasedona
realisticmacroscopic-microscopic model. Comparedtoothercalculations: (1)weuseadeformation
spaceof a sufficiently high dimension, sampled densely enough todescribe therelevanttopography
ofthefissionpotential,(2)weunambiguouslyfindthephysicallyrelevantsaddlepointsinthisspace,
and(3)weformulateourmodelsothatweobtaincontinuityofthepotentialenergydatthedivision
pointbetweenasinglesystemandseparated fissionfragmentsorcollidingnuclei,allowing usto(4)
describe both fission-barrier heightsand ground-statemasses throughout thePeriodic Table.
4 PACSnumbers: 24.75.+i,25.85.-w,21.10.Dr,25.70.Gh,21.60.Cs,21.10.Sf
0
0
2 Ithasbeennotoriouslydifficulttocalculateinaconsis- for different nuclear shapes. Such a calculation defines
tenttheoreticalmodelwithmicroscopiccontentbothfis- an energy landscape as a function of a number of shape
n
sionbarriersandground-statemassesfornucleithrough- coordinates. The fission-barrier height is given by the
a
J out the Periodic Table. So far this has only been energyrelative to the groundstate of the mostfavorable
7 possible in the framework of a macroscopic-microscopic saddle point that has to be traversed when the shape
2 model [1, 2]. However, developments in the area of nu- evolves from a single shape to separated fragments. We
clear fission show that these earlier studies can now be use a technique borrowed from the area of geographical

1 improved. topography studies, namely immersion (“imaginary wa-
v
Ontheexperimentalsideitbecameclearthatthe bar- ter flow”) [12, 13, 14], to determine the structure of the
8
5 rierdataforthetwolightestnucleipreviouslyusedinthe high-dimensional fission potential-energy surfaces [8, 9].
0 determination of model constants are incorrect. In addi- A number of models exist for the nuclear potential
1 tion, a series of measurements of fission-barrier heights energy. At first sight, it would seem attractive to em-
0 of nuclei with atomic numbers in the neighborhood of ployaself-consistentmean-field(SMF)modelusingeffec-
4 A = 100 are now available [3, 4, 5]. Barrier heights cal- tive forces, for example a Hartree-Fock (HF) or Hartree-
0
culatedfortheselightnucleiusingthemodelfromRef.[1] Fock-Bogolyubov (HFB) model with Skyrme or Gogny
/
h are from 1 to 5 MeV too low [6, 7]. effective interactions [15, 16, 17], or a relativistic Dirac-
-t On the theoretical side we have recently shown that Hartree model with scalar and vector interactions [18].
cl five-dimensional deformation spaces with the potential Global HF mass calculations have recently been pre-
u energy defined on millions of points are necessary to sented [19, 20]. However, at least two major problems
n determine properly the details of the potential energy with such calculations remain unresolved. First, no ef-
v: such as the locations and heights of the fission saddle fective force hasbeen found whichcandescribe both nu-
i points [8, 9]. This large deformation space is in stark clear masses and fission barriers for nuclei of all mass
X
contrast to one with three degrees of freedom and 175 numbers. Second, even if an appropriate effective in-
r deformation points previously used to determine the lo- teraction could be determined, it is extremely difficult,
a
cations and heights of saddle points [1, 2]. Moreover, it and in practice has so far proven impossible unambigu-
has been clear for some time that the Wigner and A0 ously to locate an actual saddle-point configuration in
macroscopic terms in these nuclear mass models must SMFmodels. Thereexistsacommonmisconceptionthat
haveashapedependence[10]. Thiswasignoredinprevi- constrained self-consistent HF or HFB calculations with
ous global calculations [1, 2, 6, 7] and only incorporated Skyrme or Gogny forces automatically take into account
in some of our more limited fission-barrier studies, for all non-constrained shape variables in a proper manner.
example [11]. In fact, the apparent saddle points that appear in con-
strained HF calculations in the general case have no re-
In nuclear fission the nucleus evolves from a single
lation to the true saddle points; we give an illustrative
ground-state shape into two separatedfission fragments.
example below.
During the shape and configuration changes that occur
in this process the total energy of the system initially For calculating the fission potential-energy surface in
increases up to a maximum, the fission-barrier height, this work, we adopt the macroscopic-microscopic finite-
then decreases. Calculations of fission barriers involve range liquid-dropmodel (FRLDM) [2] generalizedto ac-
the determination of the total nuclear potential energy count for all required shape-dependencies of its various
macroscopic terms. In contrast the finite-range droplet
model (FRDM) [2] cannot be generalizedin this way. In
Fig. 1 we illustrate the large discontinuities that occur
∗Electronicaddress: [email protected] atthe transitionpointbetweensingleandseparatedsys-
2
aslargeas5MeVevenforshapeswithafairlylargeneck
25 78Zn + 208Hg → 286110 radius. We use the three-quadratic-surface (3QS) shape
parameterization[9,21]todescribeshapeswiththesefive
Total fusion
degreesoffreedom. Byinvestigatingthescaleoverwhich
20 Macroscopic
fusion (a) the microscopic energy varies significantly, we determine
) the coordinate mesh upon which we need to define the
V
e 15 energy. We find that we get a reasonablecoverageof the
M (b) space by defining a grid of 15 points each in the neck
(
y diameter and left and right fragment deformations, 20
g
er 10 (a) points in the mass asymmetry, and 41 points in the nu-
En clear elongation. A few grid points do not refer to real
al 5 shapes, so we are left with a grid of 2,610,885points [9].
nti In an unconstrained mean-field calculation, one starts
e (b)
ot T withsomeinitialdensity,usuallydefinedintermsoftrial
P 0 ou wavefunctions,thendeterminesnewwavefunctionswhich
c
h
in are solutions of the potential derived from the density.
g
D By iterating to convergence one finds a local minimum:
− 5 ista the nuclear ground state or possibly a fission-isomeric
(a): W, A0 shape-dependent nc state. To try to find a fission barrier, some have cho-
−10 (b): W, A0 shape-independent e sen to solve a constrained problem, which leads to the
0.5 1.0 1.5 2.0 minimum-energy state subject to the constraint, often
taken to be the quadrupole moment. By applying a se-
Distance between Mass Centers r (Units of R )
0
riesofconstraintswithincreasingdeformation,acurveof
energyasafunctionofconstraineddeformationisfound.
FIG. 1: Calculated macroscopic and total potential energies,
Such curves often exhibit discontinuities and may not
for shape sequencesleading to the touchingconfiguration, at
thelong-dashedline,ofspherical78Znand208Hg. Totheleft pass through the real saddle point in multidimensional
the calculations trace the energy for a single, joined shape space as is discussed in more detail in Refs. [8, 22].
configurationfrom oblateshapesthroughthesphericalshape Inamacroscopic-microscopiccalculationoneshouldin
at r =0.75 to the touching configuration at r =1.52; to the principlebeabletolocatesaddlepointsbysolvingforall
right the calculations trace the energy for separated spheri- shapes that have a zero derivative of the energy with re-
cal nuclei beyond the touching point. The continuous path
spect to all the degrees of freedom. This method works
through five-dimensional space from the ground state to the
for a purely macroscopic model [7], but macroscopic-
touching configuration is arbitrary; the key point is that the
microscopic models using the Strutinsky shell-correction
limiting shapes when approaching the line of touching from
the left and right are identical, namely spherical 78Zn and technique[23,24],aresubjecttofluctuationswhensmall
208Hgincontact. Ataspecificvalueofr allcurvesarecalcu- shapechangesaremade,makingitdifficult toobtainac-
lated for thesame shape. To obtain continuity of themacro- curate derivatives by numerical techniques. Even if all
scopicenergyattouching,acrucialfeatureinrealisticmodels, saddlepointsinahigh-dimensionalspacecouldbefound
it isessential thatvariousmodel termsdependappropriately in this way, one must still understand the topography
onnuclearshape,as isthecaseforthecurves(a). Theslight and deduce which saddle would correspond to the ac-
remainingdiscontinuityinthetotalfusionenergycurvearises tual peak of the barrier. Before the breakthrough study
becausetheFermisurfacesofthenucleireadjustattouching, in Ref. [13], what has usually been done in calculations
and because pairing and spin-orbit strength parameters also
involving more than two degrees of freedom is to first
undergo small discontinuouschanges there.
define a two-dimensionalspace of two primary shape co-
ordinates. For each point in this two-dimensional space
the energy is then minimized with respect to a set of
additional shape degrees of freedom. It was incorrectly
tems whenthe shape dependences of the Wigner andA0 assumed that if no discontinuities occurred in this two-
macroscopic energies are neglected, and the continuous dimensionalsurfacethenitssaddlepointswouldbeiden-
behavior exhibited for the current formulation. tical to the saddle-points in the full, higher-dimensional
In the macroscopic-microscopic approach, it is impor- space. For all but the most structureless functions this
tant that the shape description be flexible enough to al- procedure is incorrect and may actually result in more
low accessing those configurations which are physically inaccurate saddle points than if only the original two-
importantto the fissionprocess. Inaddition to the com- dimensional space is studied.
monlyusedelongation,neckingandmass-asymmetryde- We illustrate in Fig. 2 some of the problems that one
grees of freedom, it is essential to include the deforma- mayencounterineitheraconstrainedSMFcalculationor
tions of the partially formed fragments. This is because in a macroscopic-microscopicmodel when attempting to
the microscopic binding due to fragment shell structure, reduce the dimensionality ofa problemvia minimization
which is sensitive to the fragment deformations, can be while preserving the essential features of the potential
3
Saddle Search Strategies Illustrated 50
10
40
30
5
V) 20
e
M
α 0 ht ( 10 Experimental
g 50/0
ei
H
− 5 er 40
− 1−0 100 −50 0 50 10Graphics by Peter Möller0 on-Barri 2300
Θ Fissi 10
Calculated (FRLDM (2002))
FIG.2: Maxima(+),minima(−),andsaddlepoints(arrows 10/0
or crossed lines) of a two-dimensional function. As discussed Discrepancy (Exp. − Calc.)
in the text it is not possible to obtain a lower-dimensional 0
representationofthissurfaceby“minimizing”withrespectto rms = 1.00 MeV
the“additional” (α) shape degree of freedom. Darkershades − 10
0 20 40 60 80 100 120 140 160
of gray indicate higher function values. Alternate contour
bandsare light gray for readability. Neutron Number N
FIG. 3: Comparison of calculated and experimental fission-
barrierheightsfornucleithroughoutthePeriodicTable,after
a readjustment of the macroscopic model constants. Exper-
energy. We assume the Θ coordinate corresponds to the imental barriers are well reproduced by the calculations, the
fission direction and α to all other degrees of freedom. rmserror isonly 0.999 MeV for 31 nuclei. Intheactinidere-
Because of the multiple saddle points, it is not clear a gion it is theouterof thetwo peaksin the“double-humped”
priori which one would correspond to the fission thresh- barrier that is compared to experimentaldata.
old. We identify the point Θ = −100, α = 6 as the
ground state or fission-isomer state and proceed to find
a “constrained” fission barrier. From the starting point
we increase Θ by 40 (smaller steps will not alter the re- uptheground-stateminimumwith“water”,determining
sult) while keeping α fixed. From the new position we when a prespecified point in the fission valley becomes
then minimize with respect to α and find ourselves at “wet”. By adjusting the increase in the water level care-
the first black dot. When we repeat this process we ob- fully we are able unambiguously to identify the location
tain the dot-dashed curve. The energy along this path and energy of the grid point nearest to the true saddle.
is a continuous function and the white arrow would be Because we are studying a higher dimensional defor-
identified as the fission saddle point. However, this sad- mation space with over 10000 times as many points as
dle is higher than those shown by gray arrows, which in previous calculations, we find that the saddles for a
can only be identified when the full space is explored. given nucleus are always lower than those found for the
Of course in a constrained SMF calculation the conver- samemodel parametersinourearlierstudies [2, 8]. This
gence towards a solution is more complex than “sliding means we need to redetermine the parameters of our
downhill”,sincethe wavefunctionsandpotentialchange
during this process. However, solutions of constrained
SMF equations do show similar behavior; often converg-
TABLE I: Macroscopic model parameters of the FRLDM
ing to a local minimum which depends on the starting
(1992) and those obtained in the present adjustment, desig-
configuration, a process similar to what is sketched in
nated FRLDM (2002) using barrier heights obtained in our
Fig. 2.
five-dimensionalcalculation.
The fundamental point is that the fission saddle point Constant FRLDM(1992) FRLDM(2002)
can only be determined from global properties of the av 16.00126 MeV 16.02500 MeV
multidimensional energy surface, not from local excur- κv 1.92240 MeV 1.93200 MeV
sions from a specific starting point. We therefore imple- as 21.18466 MeV 21.33000 MeV
ment the immersion method mentioned above and first κs 2.34500 MeV 2.37800 MeV
identify all minima by locating the points which have a0 2.61500 MeV 2.04000 MeV
a lower energy than all 3n−1 neighboring points in n- ca 0.10289 MeV 0.09700 MeV
dimensional coordinate space. We then progressively fill
4
the creation of our 1995 mass table [2]. We show in Ta-
TABLEII: Calculated barriersfor31nucleicomparedtoex-
ble I the constants from Ref. [2] (determined in 1992)
perimentaldata. Thefirst5barriersaremacroscopicbarriers.
and those obtained in our readjustment taking into ac-
Z A Eexp Ecalc ∆E Z A Eexp Ecalc ∆E countthelargerdeformationspaceinlocatingthe fission
saddlepoints. Wehavemadeanumberoftestswhichal-
(MeV) (MeV) (MeV) (MeV) (MeV) (MeV)
low us to conclude thata self-consistentredetermination
34 70 39.40 37.58 1.81 92 238 5.50 5.48 0.01
oftheground-stateandsaddle-pointdeformationswould
34 76 44.50 43.84 0.65 92 240 5.50 6.27 −0.77
change our calculated energies by less than 0.1 MeV.
42 90 40.92 40.92 −0.00 94 236 4.50 4.35 0.14
42 94 44.68 44.20 0.47 94 238 5.00 4.39 0.60 These constants (in particular a0) differ slightly from
42 98 45.84 46.88 −1.04 94 240 5.15 4.83 0.31 the preliminary set presented in Ref. [25]. Those con-
80 198 20.40 21.41 −1.01 94 242 5.05 5.55 −0.50 stants were affected by an error in the expression for
84 210 21.40 22.02 −0.62 94 244 5.00 6.29 −1.29 the a0 energy term in the constant-adjustmentprogram.
84 212 19.50 20.20 −0.70 94 246 5.30 7.01 −1.71
However, none of the other previous results, conclusions
88 228 8.10 7.45 0.64 96 242 5.00 4.28 0.71
or figures were affected significantly by this computer-
90 228 6.50 6.47 0.02 96 244 5.10 5.02 0.07
90 230 7.00 5.65 1.34 96 246 4.80 5.81 −1.01 program bug. Here we have checked the calculation of
90 232 6.20 5.45 0.74 96 248 4.80 6.41 −1.61 macroscopic-model saddle-point shapes and energies by
90 234 6.50 5.36 1.13 96 250 4.40 5.98 −1.58 the use of two independently written codes. The micro-
92 232 5.40 4.67 0.72 98 250 3.60 5.88 −2.28 scopic energy model is unchanged from [2].
92 234 5.50 4.89 0.60 98 252 4.80 5.63 −0.83
The1992calculationreproducedanexperimental1989
92 236 5.67 4.98 0.68
nuclearmasstable[26]withamodelerrorof0.779MeV,
and28barrierheightswith model errorof1.4MeV. The
revised data set here [4, 5, 27] incorporates seven new
experimental barrier heights and removes four old ones.
nuclear-structure model. Because new parameter sets
Thefittotherevisedtableof31barriershasanrmserror
maychangethelocationofthesaddlepointsandminima,
of 0.999 MeV, and the fit to the same 1989 mass table
an iterative procedure is in principle required. However
has a model error of 0.752 MeV using the parameters
these changes in deformation are small; here we do just
in the last column of Table I. We show the experimen-
the first iteration and vary six parameters of the macro-
tal and calculated barrier heights as well as remaining
scopic energy functional to optimize the reproduction
discrepancies in Table II and Fig. 3.
of both ground-state binding energies and fission-barrier
heights. This process is identical to the one followed in This work was supported by the U. S. DOE.
[1] P.M¨ollerandJ.R.Nix,AtomicDataNucl.DataTables [13] A.Mamdouh,J.M.Pearson,M.Rayet,andF.Tondeur,
26. 165 (1981). Nucl. Phys. A644, 389 (1998).
[2] P.M¨oller, J. R.Nix,W. D.Myers, and W. J. Swiatecki, [14] B. Hayes,Am. Sci. 88, 481 (2000).
AtomicData Nucl. Data Tables 59, 185 (1995). [15] S. ˚Aberg, H. Flocard, and W. Nazarewicz, Ann. Rev.
[3] M. A. McMahan, L. G. Moretto, M. L. Padgett, G. J. Nucl. Sci. 40, 439 (1990).
Wozniak,L.G.Sobotka,andM.G.Mustafa, Phys.Rev. [16] J. L. Egido, L. M. Robledo, and R. R. Chasman, Phys.
Lett.54, 1995 (1985). Lett. B393, 13 (1997).
[4] D.N.Delis,Y.Blumenfeld,D.R.Bowman, N.Colonna, [17] J.F.Berger,M.Girod,andD.Gogny,Nucl.Phys.A502,
K.Hanold,K.Jing,M.Justice,J.C.Meng,G.F.Peaslee, c85 (1989).
G. J. Wozniak, and L. G. Moretto, Nucl. Phys A 534, [18] B.A.Nikolaus,T.Hoch,andD.G.Madland,Phys.Rev.
403 (1991). C 46, 1757 (1992).
[5] K. X. Jing, L. G. Moretto, A. C. Veeck, N. Colonna, I. [19] S.Goriely,F.Tondeur,andJ.M.Pearson,At.DataNucl
Lhenry, K. Tso, K. Hanold, W. Skulski, Q. Sui, and G. Data Tables, 77, 311 (2001).
J. Wozniak, Nucl. Phys.A645, 203 (1999). [20] S.Goriely, M.Samyn,P.-H.Heenen,J. M. Pearson, and
[6] A.J. Sierk,Phys. Rev.Lett. 55, 582 (1985). F. TondeurPhys.Rev.C 66, 024326 (2002).
[7] A.J. Sierk,Phys. Rev.C 33, 2039 (1986). [21] J. R.Nix, Nucl.Phys. A130, 241 (1969).
[8] P. M¨oller and A. Iwamoto, Phys. Rev. C 61, 047602 [22] W. D. Myers and W. J. Swiatecki, Nucl. Phys. A601
(2000). (1996) 141.
[9] P. M¨oller, D. G. Madland, A. J. Sierk, and A. Iwamoto, [23] V. M. Strutinsky,Nucl. Phys. A95, 420 (1967).
Nature409, 785 (2001). [24] V. M. Strutinsky,Nucl. Phys. A122, 1 (1968).
[10] W. D. Myers, Droplet model of atomic nuclei [25] Peter M¨oller, D. G. Madland, A. J. Sierk, and A.
(IFI/Plenum,New York,1977). Iwamoto, Proc. International Conference on Nuclear
[11] P. M¨oller, J. R. Nix, and W. J. Swiatecki, Nucl. Phys. Data for Science and Technology (ND2001), October
A492, 349 (1989). 7–12, Tsukuba, Japan, Journal of Nuclear Science and
[12] V.LucandP.Soille,IEEETransactionsonPatternAna- Technology, Supplement2, 703 (2002).
lyis and Machine Intelligence, 13, 583 (1991). [26] G. Audi, Midstream atomic mass evaluation, private
5
communication (1989), with four revisions.
[27] D.G. Madland, Private communication (2001).